Opposition from the Senate's top Democrat to the White House's trade agenda has highlighted a broader reality: The quest for new overseas deals has a diminishing number of friends in Congress.

In the 12 years since the legislature last granted a president special trade powers, Capitol Hill has changed significantly. Republicans, especially many tea-party-backed newcomers, are increasingly leery on the trade front and reluctant to grant President Barack Obama negotiating powers known as fast track. The Senate also has lost...